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[libreriscv.git] / openpower / sv / rfc / ls009.mdwn

# RFC ls009 Simple-V REMAP Subsystem

**URLs**:

* <https://libre-soc.org/openpower/sv/>
* <https://libre-soc.org/openpower/sv/rfc/ls009/>
* <https://bugs.libre-soc.org/show_bug.cgi?id=1042>
* <https://git.openpower.foundation/isa/PowerISA/issues/87>

**Severity**: Major

**Status**: New

**Date**: 26 Mar 2023

**Target**: v3.2B

**Source**: v3.0B

**Books and Section affected**:

```
    Book I, new Zero-Overhead-Loop Chapter.
    Appendix E Power ISA sorted by opcode
    Appendix F Power ISA sorted by version
    Appendix G Power ISA sorted by Compliancy Subset
    Appendix H Power ISA sorted by mnemonic
```

**Summary**

```
    svremap  - Re-Mapping of Register Element Offsets
    svindex  - General-purpose setting of SHAPEs to be re-mapped
    svshape  - Hardware-level setting of SHAPEs for element re-mapping
    svshape2 - Hardware-level setting of SHAPEs for element re-mapping (v2)
```

**Submitter**: Luke Leighton (Libre-SOC)

**Requester**: Libre-SOC

**Impact on processor**:

```
    Addition of four new "Zero-Overhead-Loop-Control" DSP-style Vector-style
    Management Instructions which provide advanced features such as Matrix
    FFT DCT Hardware-Assist Schedules and general-purpose Index reordering.
```

**Impact on software**:

```
    Requires support for new instructions in assembler, debuggers,
    and related tools.
```

**Keywords**:

```
    Cray Supercomputing, Vectorisation, Zero-Overhead-Loop-Control (ZOLC),
    Scalable Vectors, Multi-Issue Out-of-Order, Sequential Programming Model,
    Digital Signal Processing (DSP)
```

**Motivation**

These REMAP Management instructions provide state-of-the-art advanced capabilities
to dramatically decrease instruction count and power reduction whilst retaining
unprecedented general-purpose capability and a standard Sequential Execution Model.

**Notes and Observations**:

1. TODO

**Changes**

Add the following entries to:

* the Appendices of Book I
* Instructions of Book I as a new Section
* TODO-Form of Book I Section 1.6.1.6 and 1.6.2

----------------

\newpage{}

# REMAP <a name="remap" />

REMAP is an advanced form of Vector "Structure Packing" that
provides hardware-level support for commonly-used *nested* loop patterns
that would otherwise require full inline loop unrolling.
For more general reordering an Indexed REMAP mode is available
(an abstracted analog to `xxperm`).

REMAP allows the usual sequential vector loop `0..VL-1` to be "reshaped" (re-mapped)
from a linear form to a 2D or 3D transposed form, or "offset" to permit
arbitrary access to elements (when elwidth overrides are used),
independently on each Vector src or dest
register. Aside from Indexed REMAP this is entirely Hardware-accelerated
reordering and consequently not costly in terms of register access. It
will however place a burden on Multi-Issue systems but no more than if
the equivalent Scalar instructions were explicitly
loop-unrolled without SVP64, and some advanced implementations may even find
the Deterministic nature of the Scheduling to be easier on resources.

The initial primary motivation of REMAP was for Matrix Multiplication, reordering
of sequential data in-place: in-place DCT and FFT were easily justified given the
exceptionally high usage in Computer Science.
Four SPRs are provided which may be applied to any GPR, FPR or CR Field
so that for example a single FMAC may be
used in a single hardware-controlled 100% Deterministic loop to
perform 5x3 times 3x4 Matrix multiplication,
generating 60 FMACs *without needing explicit assembler unrolling*.
Additional uses include regular "Structure Packing"
such as RGB pixel data extraction and reforming.

REMAP, like all of SV, is abstracted out, meaning that unlike traditional
Vector ISAs which would typically only have a limited set of instructions
that can be structure-packed (LD/ST typically), REMAP may be applied to
literally any instruction: CRs, Arithmetic, Logical, LD/ST, anything.

Note that REMAP does not *directly* apply to sub-vector elements but
only to the group: that 
is what swizzle is for.  Swizzle *can* however be applied to the same
instruction as REMAP.  As explained in [[sv/mv.swizzle]]
and the [[svp64/appendix]], Pack and Unpack EXTRA Mode bits
can extend down into Sub-vector elements to perform vec2/vec3/vec4
sequential reordering, but even here, REMAP is not *individually*
extended down to the actual sub-vector elements themselves.

In its general form, REMAP is quite expensive to set up, and on some
implementations may introduce
latency, so should realistically be used only where it is worthwhile.
Given that even with latency the fact that up to 127 operations
can be requested to be issued (from a single instruction) it should
be clear that REMAP should not be dismissed for *possible* latency alone.
Commonly-used patterns such as Matrix Multiply, DCT and FFT have
helper instruction options which make REMAP easier to use.

There are four types of REMAP:

* **Matrix**, also known as 2D and 3D reshaping, can perform in-place
  Matrix transpose and rotate. The Shapes are set up for an "Outer Product"
  Matrix Multiply.
* **FFT/DCT**, with full triple-loop in-place support: limited to
  Power-2 RADIX
* **Indexing**, for any general-purpose reordering, also includes
  limited 2D reshaping.
* **Parallel Reduction**, for scheduling a sequence of operations
  in a Deterministic fashion, in a way that may be parallelised,
  to reduce a Vector down to a single value.

Best implemented on top of a Multi-Issue Out-of-Order Micro-architecture,
REMAP Schedules are 100% Deterministic **including Indexing** and are
designed to be incorporated in between the Decode and Issue phases,
directly into Register Hazard Management.

Parallel Reduction is unusual in that it requires a full vector array
of results (not a scalar) and uses the rest of the result Vector for
the purposes of storing intermediary calculations.  As these intermediary
results are Deterministically computed they may be useful.
Additionally, because the intermediate results are always written out
it is possible to service Precise Interrupts without affecting latency
(a common limitation of Vector ISAs implementing explicit
Parallel Reduction instructions).

## Basic principle

* normal vector element read/write of operands would be sequential
  (0 1 2 3 ....)
* this is not appropriate for (e.g.) Matrix multiply which requires
  accessing elements in alternative sequences (0 3 6 1 4 7 ...)
* normal Vector ISAs use either Indexed-MV or Indexed-LD/ST to "cope"
  with this.  both are expensive (copy large vectors, spill through memory)
  and very few Packed SIMD ISAs cope with non-Power-2.
* REMAP **redefines** the order of access according to set
  (Deterministic) "Schedules".
* The Schedules are not at all restricted to power-of-two boundaries
  making it unnecessary to have for example specialised 3x4 transpose
  instructions of other Vector ISAs.

Only the most commonly-used algorithms in computer science have REMAP
support, due to the high cost in both the ISA and in hardware.  For
arbitrary remapping the `Indexed` REMAP may be used.

## Example Usage

* `svshape` to set the type of reordering to be applied to an
  otherwise usual `0..VL-1` hardware for-loop
* `svremap` to set which registers a given reordering is to apply to
  (RA, RT etc)
* `sv.{instruction}` where any Vectorised register marked by `svremap`
  will have its ordering REMAPPED according to the schedule set
  by `svshape`.

The following illustrative example multiplies a 3x4 and a 5x3
matrix to create
a 5x4 result:

```
    svshape 5, 4, 3, 0, 0
    svremap 15, 1, 2, 3, 0, 0, 0, 0
    sv.fmadds *0, *8, *16, *0
```

* svshape sets up the four SVSHAPE SPRS for a Matrix Schedule
* svremap activates four out of five registers RA RB RC RT RS (15)
* svremap requests:
  - RA to use SVSHAPE1
  - RB to use SVSHAPE2
  - RC to use SVSHAPE3
  - RT to use SVSHAPE0
  - RS Remapping to not be activated
* sv.fmadds has RT=0.v, RA=8.v, RB=16.v, RC=0.v
* With REMAP being active each register's element index is
  *independently* transformed using the specified SHAPEs.

Thus the Vector Loop is arranged such that the use of
the multiply-and-accumulate instruction executes precisely the required
Schedule to perform an in-place in-registers Matrix Multiply with no
need to perform additional Transpose or register copy instructions.
The example above may be executed as a unit test and demo,
[here](https://git.libre-soc.org/?p=openpower-isa.git;a=blob;f=src/openpower/decoder/isa/test_caller_svp64_matrix.py;h=c15479db9a36055166b6b023c7495f9ca3637333;hb=a17a252e474d5d5bf34026c25a19682e3f2015c3#l94)
     
## REMAP types

This section summarises the motivation for each REMAP Schedule
and briefly goes over their characteristics and limitations.
Further details on the Deterministic Precise-Interruptible algorithms
used in these Schedules is found in the [[sv/remap/appendix]].

### Matrix (1D/2D/3D shaping)

Matrix Multiplication is a huge part of High-Performance Compute,
and 3D.
In many PackedSIMD as well as Scalable Vector ISAs, non-power-of-two
Matrix sizes are a serious challenge. PackedSIMD ISAs, in order to
cope with for example 3x4 Matrices, recommend rolling data-repetition and loop-unrolling.
Aside from the cost of the load on the L1 I-Cache, the trick only
works if one of the dimensions X or Y are power-two. Prime Numbers
(5x7, 3x5) become deeply problematic to unroll.

Even traditional Scalable Vector ISAs have issues with Matrices, often
having to perform data Transpose by pushing out through Memory and back,
or computing Transposition Indices (costly) then copying to another
Vector (costly).

Matrix REMAP was thus designed to solve these issues by providing Hardware
Assisted
"Schedules" that can view what would otherwise be limited to a strictly
linear Vector as instead being 2D (even 3D) *in-place* reordered.
With both Transposition and non-power-two being supported the issues
faced by other ISAs are mitigated.

Limitations of Matrix REMAP are that the Vector Length (VL) is currently
restricted to 127: up to 127 FMAs (or other operation)
may be performed in total.
Also given that it is in-registers only at present some care has to be
taken on regfile resource utilisation. However it is perfectly possible
to utilise Matrix REMAP to perform the three inner-most "kernel" loops of
the usual 6-level large Matrix Multiply, without the usual difficulties
associated with SIMD.

Also the `svshape` instruction only provides access to part of the
Matrix REMAP capability. Rotation and mirroring need to be done by
programming the SVSHAPE SPRs directly, which can take a lot more
instructions. Future versions of SVP64 will include EXT1xx prefixed
variants (`psvshape`) which provide more comprehensive capacity and
mitigate the need to write direct to the SVSHAPE SPRs.

### FFT/DCT Triple Loop

DCT and FFT are some of the most astonishingly used algorithms in
Computer Science.  Radar, Audio, Video, R.F. Baseband and dozens more.  At least
two DSPs, TMS320 and Hexagon, have VLIW instructions specially tailored
to FFT.

An in-depth analysis showed that it is possible to do in-place in-register
DCT and FFT as long as twin-result "butterfly" instructions are provided.
These can be found in the [[openpower/isa/svfparith]] page if performing
IEEE754 FP transforms. *(For fixed-point transforms, equivalent 3-in 2-out
integer operations would be required)*. These "butterfly" instructions
avoid the need for a temporary register because the two array positions
being overwritten will be "in-flight" in any In-Order or Out-of-Order
micro-architecture.

DCT and FFT Schedules are currently limited to RADIX2 sizes and do not
accept predicate masks.  Given that it is common to perform recursive
convolutions combining smaller Power-2 DCT/FFT to create larger DCT/FFTs
in practice the RADIX2 limit is not a problem.  A Bluestein convolution
to compute arbitrary length is demonstrated by
[Project Nayuki](https://www.nayuki.io/res/free-small-fft-in-multiple-languages/fft.py)

### Indexed

The purpose of Indexing is to provide a generalised version of
Vector ISA "Permute" instructions, such as VSX `vperm`.  The
Indexing is abstracted out and may be applied to much more
than an element move/copy, and is not limited for example
to the number of bytes that can fit into a VSX register.
Indexing may be applied to LD/ST (even on Indexed LD/ST
instructions such as `sv.lbzx`), arithmetic operations,
extsw: there is no artificial limit.

The only major caveat is that the registers to be used as
Indices must not be modified by any instruction after Indexed Mode
is established, and neither must MAXVL be altered. Additionally,
no register used as an Index may exceed MAXVL-1.

Failure to observe
these conditions results in `UNDEFINED` behaviour.
These conditions allow a Read-After-Write (RAW) Hazard to be created on
the entire range of Indices to be subsequently used, but a corresponding
Write-After-Read Hazard by any instruction that modifies the Indices
**does not have to be created**. Given the large number of registers
involved in Indexing this is a huge resource saving and reduction
in micro-architectural complexity. MAXVL is likewise
included in the RAW Hazards because it is involved in calculating
how many registers are to be considered Indices.

With these Hazard Mitigations in place, high-performance implementations
may read-cache the Indices at the point where a given `svindex` instruction
is called (or SVSHAPE SPRs - and MAXVL - directly altered) by issuing
background GPR register file reads whilst other instructions are being
issued and executed.

The original motivation for Indexed REMAP was to mitigate the need to add
an expensive `mv.x` to the Scalar ISA, which was likely to be rejected as
a stand-alone instruction.  Usually a Vector ISA would add a non-conflicting
variant (as in VSX `vperm`) but it is common to need to permute by source,
with the risk of conflict, that has to be resolved, for example, in AVX-512
with `conflictd`.

Indexed REMAP on the other hand **does not prevent conflicts** (overlapping
destinations), which on a superficial analysis may be perceived to be a
problem, until it is recalled that, firstly, Simple-V is designed specifically
to require Program Order to be respected, and that Matrix, DCT and FFT
all *already* critically depend on overlapping Reads/Writes: Matrix
uses overlapping registers as accumulators.  Thus the Register Hazard
Management needed by Indexed REMAP *has* to be in place anyway.

The cost compared to Matrix and other REMAPs (and Pack/Unpack) is
clearly that of the additional reading of the GPRs to be used as Indices,
plus the setup cost associated with creating those same Indices.
If any Deterministic REMAP can cover the required task, clearly it
is adviseable to use it instead.

*Programmer's note: some algorithms may require skipping of Indices exceeding
VL-1, not MAXVL-1. This may be achieved programmatically by performing
an `sv.cmp *BF,*RA,RB` where RA is the same GPRs used in the Indexed REMAP,
and RB contains the value of VL returned from `setvl`. The resultant
CR Fields may then be used as Predicate Masks to exclude those operations
with an Index exceeding VL-1.*

### Parallel Reduction

Vector Reduce Mode issues a deterministic tree-reduction schedule to the underlying micro-architecture.  Like Scalar reduction, the "Scalar Base"
(Power ISA v3.0B) operation is leveraged, unmodified, to give the
*appearance* and *effect* of Reduction.

In Horizontal-First Mode, Vector-result reduction **requires**
the destination to be a Vector, which will be used to store
intermediary results.

Given that the tree-reduction schedule is deterministic,
Interrupts and exceptions
can therefore also be precise.  The final result will be in the first
non-predicate-masked-out destination element, but due again to
the deterministic schedule programmers may find uses for the intermediate
results.

When Rc=1 a corresponding Vector of co-resultant CRs is also
created.  No special action is taken: the result *and its CR Field*
are stored "as usual" exactly as all other SVP64 Rc=1 operations.

Note that the Schedule only makes sense on top of certain instructions:
X-Form with a Register Profile of `RT,RA,RB` is fine because two sources
and the destination are all the same type.  Like Scalar
Reduction, nothing is prohibited:
the results of execution on an unsuitable instruction may simply
not make sense. With care, even 3-input instructions (madd, fmadd, ternlogi) 
may be used, and whilst it is down to the Programmer to walk through the
process the Programmer can be confident that the Parallel-Reduction is
guaranteed 100% Deterministic.

Critical to note regarding use of Parallel-Reduction REMAP is that,
exactly as with all REMAP Modes, the `svshape` instruction *requests*
a certain Vector Length (number of elements to reduce) and then
sets VL and MAXVL at the number of **operations** needed to be
carried out.  Thus, equally as importantly, like Matrix REMAP
the total number of operations
is restricted to 127.  Any Parallel-Reduction requiring more operations
will need to be done manually in batches (hierarchical
recursive Reduction).

Also important to note is that the Deterministic Schedule is arranged
so that some implementations *may* parallelise it (as long as doing so
respects Program Order and Register Hazards).  Performance (speed)
of any given
implementation is neither strictly defined or guaranteed.  As with
the Vulkan(tm) Specification, strict compliance is paramount whilst
performance is at the discretion of Implementors.

**Parallel-Reduction with Predication**

To avoid breaking the strict RISC-paradigm, keeping the Issue-Schedule
completely separate from the actual element-level (scalar) operations,
Move operations are **not** included in the Schedule.  This means that
the Schedule leaves the final (scalar) result in the first-non-masked 
element of the Vector used.  With the predicate mask being dynamic
(but deterministic) this result could be anywhere.

If that result is needed to be moved to a (single) scalar register
then a follow-up `sv.mv/sm=predicate rt, *ra` instruction will be
needed to get it, where the predicate is the exact same predicate used
in the prior Parallel-Reduction instruction.

* If there was only a single
  bit in the predicate then the result will not have moved or been altered
  from the source vector prior to the Reduction
* If there was more than one bit the result will be in the
  first element with a predicate bit set.

In either case the result is in the element with the first bit set in
the predicate mask. Thus, no move/copy *within the Reduction itself* was needed.

Programmer's Note: For *some* hardware implementations
the vector-to-scalar copy may be a slow operation, as may the Predicated
Parallel Reduction itself.
It may be better to perform a pre-copy
of the values, compressing them (VREDUCE-style) into a contiguous block,
which will guarantee that the result goes into the very first element
of the destination vector, in which case clearly no follow-up
predicated vector-to-scalar MV operation is needed.

**Usage conditions**

The simplest usage is to perform an overwrite, specifying all three
register operands the same.

```
    svshape parallelreduce, 6
    sv.add *8, *8, *8
```

The Reduction Schedule will issue the Parallel Tree Reduction spanning
registers 8 through 13, by adjusting the offsets to RT, RA and RB as
necessary (see "Parallel Reduction algorithm" in a later section).

A non-overwrite is possible as well but just as with the overwrite
version, only those destination elements necessary for storing
intermediary computations will be written to: the remaining elements
will **not** be overwritten and will **not** be zero'd.

```
    svshape parallelreduce, 6
    sv.add *0, *8, *8
```

However it is critical to note that if the source and destination are
not the same then the trick of using a follow-up vector-scalar MV will
not work.

### Sub-Vector Horizontal Reduction

To achieve Sub-Vector Horizontal Reduction, Pack/Unpack should be enabled,
which will turn the Schedule around such that issuing of the Scalar
Defined Words is done with SUBVL looping as the inner loop not the
outer loop. Rc=1 with Sub-Vectors (SUBVL=2,3,4) is `UNDEFINED` behaviour.

## Determining Register Hazards

For high-performance (Multi-Issue, Out-of-Order) systems it is critical
to be able to statically determine the extent of Vectors in order to
allocate pre-emptive Hazard protection.  The next task is to eliminate
masked-out elements using predicate bits, freeing up the associated
Hazards.

For non-REMAP situations `VL` is sufficient to ascertain early
Hazard coverage, and with SVSTATE being a high priority cached
quantity at the same level of MSR and PC this is not a problem.

The problems come when REMAP is enabled.  Indexed REMAP must instead
use `MAXVL` as the earliest (simplest)
batch-level Hazard Reservation indicator (after taking element-width
overriding on the Index source into consideration),
but Matrix, FFT and Parallel Reduction must all use completely different
schemes.  The reason is that VL is used to step through the total
number of *operations*, not the number of registers.
The "Saving Grace" is that all of the REMAP Schedules are 100% Deterministic.

Advance-notice Parallel computation and subsequent cacheing
of all of these complex Deterministic REMAP Schedules is
*strongly recommended*, thus allowing clear and precise multi-issue
batched Hazard coverage to be deployed, *even for Indexed Mode*.
This is only possible for Indexed due to the strict guidelines
given to Programmers.

In short, there exists solutions to the problem of Hazard Management,
with varying degrees of refinement possible at correspondingly
increasing levels of complexity in hardware.

A reminder: when Rc=1 each result register (element) has an associated
co-result CR Field (one per result element).  Thus above when determining
the Write-Hazards for result registers the corresponding Write-Hazards for the
corresponding associated co-result CR Field must not be forgotten, *including* when
Predication is used.

## REMAP area of SVSTATE SPR

The following bits of the SVSTATE SPR are used for REMAP:

|32.33|34.35|36.37|38.39|40.41| 42.46 | 62 |
| --  | --  | --  | --  | --  | ----- | ------ |
|mi0  |mi1  |mi2  |mo0  |mo1  | SVme  | RMpst    |

mi0-2 and mo0-1 each select SVSHAPE0-3 to apply to a given register.
mi0-2 apply to RA, RB, RC respectively, as input registers, and
likewise mo0-1 apply to output registers (RT/FRT, RS/FRS) respectively.
SVme is 5 bits (one for each of mi0-2/mo0-1) and indicates whether the
SVSHAPE is actively applied or not.

* bit 0 of SVme indicates if mi0 is applied to RA / FRA / BA / BFA
* bit 1 of SVme indicates if mi1 is applied to RB / FRB / BB
* bit 2 of SVme indicates if mi2 is applied to RC / FRC / BC
* bit 3 of SVme indicates if mo0 is applied to RT / FRT / BT / BF
* bit 4 of SVme indicates if mo1 is applied to Effective Address / FRS / RS
  (LD/ST-with-update has an implicit 2nd write register, RA)

The "persistence" bit if set will result in all Active REMAPs being applied
indefinitely.

----------------

\newpage{}

# svremap instruction <a name="svremap"> </a>

SVRM-Form:

    svremap SVme,mi0,mi1,mi2,mo0,mo2,pst

|0     |6     |11  |13   |15   |17   |19   |21    | 22.25 |26..31 |
| --   | --   | -- | --  | --  | --  | --  | --   | ----  | ----- |
| PO   | SVme |mi0 | mi1 | mi2 | mo0 | mo1 | pst  | rsvd  | XO    |

SVRM-Form

* svremap SVme,mi0,mi1,mi2,mo0,mo1,pst

Pseudo-code:

```
    # registers RA RB RC RT EA/FRS SVSHAPE0-3 indices
    SVSTATE[32:33] <- mi0
    SVSTATE[34:35] <- mi1
    SVSTATE[36:37] <- mi2
    SVSTATE[38:39] <- mo0
    SVSTATE[40:41] <- mo1
    # enable bit for RA RB RC RT EA/FRS
    SVSTATE[42:46] <- SVme
    # persistence bit (applies to more than one instruction)
    SVSTATE[62] <- pst
```

Special Registers Altered:

```
    SVSTATE
```

`svremap` determines the relationship between registers and SVSHAPE SPRs.
The bitmask `SVme` determines which registers have a REMAP applied, and mi0-mo1
determine which shape is applied to an activated register.  the `pst` bit if
cleared indicated that the REMAP operation shall only apply to the immediately-following
instruction.  If set then REMAP remains permanently enabled until such time as it is
explicitly disabled, either by `setvl` setting a new MAXVL, or with another
`svremap` instruction. `svindex` and `svshape2` are also capable of setting or
clearing persistence, as well as partially covering a subset of the capability of
`svremap` to set register-to-SVSHAPE relationships.

Programmer's Note: applying non-persistent `svremap` to an instruction that has
no REMAP enabled or is a Scalar operation will obviously have no effect but
the bits 32 to 46 will at least have been set in SVSTATE. This may prove useful
when using `svindex` or `svshape2`.

Hardware Architectural Note: when persistence is not set it is critically important
to treat the `svremap` and the following SVP64 instruction as an indivisible fused operation.
*No state* is stored in the SVSTATE SPR in order to allow continuation should an
Interrupt occur between the two instructions. Thus, Interrupts must be prohibited
from occurring or other workaround deployed.  When persistence is set this issue
is moot.

It is critical to note that if persistence is clear then `svremap` is the *only* way
to activate REMAP on any given (following) instruction.  If persistence is set however then
**all** SVP64 instructions go through REMAP as long as `SVme` is non-zero.

-------------

\newpage{}

# SHAPE Remapping SPRs

There are four "shape" SPRs, SHAPE0-3, 32-bits in each,
which have the same format.  

Shape is 32-bits.  When SHAPE is set entirely to zeros, remapping is
disabled: the register's elements are a linear (1D) vector.

|31.30|29..28 |27..24| 23..21 | 20..18  | 17..12  |11..6 |5..0  | Mode  |
|---- |------ |------| ------ | ------- | ------- |----- |----- | ----- |
|0b00 |skip   |offset| invxyz | permute | zdimsz  |ydimsz|xdimsz|Matrix |
|0b00 |elwidth|offset|sk1/invxy|0b110/0b111|SVGPR|ydimsz|xdimsz|Indexed|
|0b01 |submode|offset| invxyz | submode2| zdimsz  |mode  |xdimsz|DCT/FFT|
|0b10 |submode|offset| invxyz | rsvd    | rsvd    |rsvd  |xdimsz|Preduce|
|0b11 |       |      |        |         |         |      |      |rsvd   |

mode sets different behaviours (straight matrix multiply, FFT, DCT).

* **mode=0b00** sets straight Matrix Mode
* **mode=0b00** with permute=0b110 or 0b111 sets Indexed Mode
* **mode=0b01** sets "FFT/DCT" mode and activates submodes
* **mode=0b10** sets "Parallel Reduction" Schedules.

## Parallel Reduction Mode

Creates the Schedules for Parallel Tree Reduction.

* **submode=0b00** selects the left operand index
* **submode=0b01** selects the right operand index

* When bit 0 of `invxyz` is set, the order of the indices
  in the inner for-loop are reversed. This has the side-effect
  of placing the final reduced result in the last-predicated element.
  It also has the indirect side-effect of swapping the source
  registers: Left-operand index numbers will always exceed
  Right-operand indices.
  When clear, the reduced result will be in the first-predicated
  element, and Left-operand indices will always be *less* than
  Right-operand ones.
* When bit 1 of `invxyz` is set, the order of the outer loop
  step is inverted: stepping begins at the nearest power-of two
  to half of the vector length and reduces by half each time.
  When clear the step will begin at 2 and double on each
  inner loop.

## FFT/DCT mode

submode2=0 is for FFT. For FFT submode the following schedules may be 
selected:

* **submode=0b00** selects the ``j`` offset of the innermost for-loop
  of Tukey-Cooley
* **submode=0b10** selects the ``j+halfsize`` offset of the innermost for-loop
  of Tukey-Cooley
* **submode=0b11** selects the ``k`` of exptable (which coefficient)

When submode2 is 1 or 2, for DCT inner butterfly submode the following
schedules may be selected.  When submode2 is 1, additional bit-reversing
is also performed.

* **submode=0b00** selects the ``j`` offset of the innermost for-loop,
    in-place
* **submode=0b010** selects the ``j+halfsize`` offset of the innermost for-loop,
  in reverse-order, in-place
* **submode=0b10** selects the ``ci`` count of the innermost for-loop,
  useful for calculating the cosine coefficient
* **submode=0b11** selects the ``size`` offset of the outermost for-loop,
  useful for the cosine coefficient ``cos(ci + 0.5) * pi / size``

When submode2 is 3 or 4, for DCT outer butterfly submode the following
schedules may be selected.  When submode is 3, additional bit-reversing
is also performed.

* **submode=0b00** selects the ``j`` offset of the innermost for-loop,
* **submode=0b01** selects the ``j+1`` offset of the innermost for-loop,

`zdimsz` is used as an in-place "Stride", particularly useful for
column-based in-place DCT/FFT.

## Matrix Mode

In Matrix Mode, skip allows dimensions to be skipped from being included
in the resultant output index.  this allows sequences to be repeated:
```0 0 0 1 1 1 2 2 2 ...``` or in the case of skip=0b11 this results in
modulo ```0 1 2 0 1 2 ...```

* **skip=0b00** indicates no dimensions to be skipped
* **skip=0b01** sets "skip 1st dimension"
* **skip=0b10** sets "skip 2nd dimension"
* **skip=0b11** sets "skip 3rd dimension"

invxyz will invert the start index of each of x, y or z. If invxyz[0] is
zero then x-dimensional counting begins from 0 and increments, otherwise
it begins from xdimsz-1 and iterates down to zero. Likewise for y and z.

offset will have the effect of offsetting the result by ```offset``` elements:

```
    for i in 0..VL-1:
        GPR(RT + remap(i) + SVSHAPE.offset) = ....
```

this appears redundant because the register RT could simply be changed by a compiler, until element width overrides are introduced.  also
bear in mind that unlike a static compiler SVSHAPE.offset may
be set dynamically at runtime.

xdimsz, ydimsz and zdimsz are offset by 1, such that a value of 0 indicates
that the array dimensionality for that dimension is 1. any dimension
not intended to be used must have its value set to 0 (dimensionality
of 1).  A value of xdimsz=2 would indicate that in the first dimension
there are 3 elements in the array.  For example, to create a 2D array
X,Y of dimensionality X=3 and Y=2, set xdimsz=2, ydimsz=1 and zdimsz=0

The format of the array is therefore as follows:

```
    array[xdimsz+1][ydimsz+1][zdimsz+1]
```

However whilst illustrative of the dimensionality, that does not take the
"permute" setting into account.  "permute" may be any one of six values
(0-5, with values of 6 and 7 indicating "Indexed" Mode).  The table
below shows how the permutation dimensionality order works:

| permute | order | array format             |
| ------- | ----- | ------------------------ |
| 000     | 0,1,2 | (xdim+1)(ydim+1)(zdim+1) |
| 001     | 0,2,1 | (xdim+1)(zdim+1)(ydim+1) |
| 010     | 1,0,2 | (ydim+1)(xdim+1)(zdim+1) |
| 011     | 1,2,0 | (ydim+1)(zdim+1)(xdim+1) |
| 100     | 2,0,1 | (zdim+1)(xdim+1)(ydim+1) |
| 101     | 2,1,0 | (zdim+1)(ydim+1)(xdim+1) |
| 110     | 0,1   | Indexed (xdim+1)(ydim+1) |
| 111     | 1,0   | Indexed (ydim+1)(xdim+1) |

In other words, the "permute" option changes the order in which
nested for-loops over the array would be done.  See executable
python reference code for further details.

*Note: permute=0b110 and permute=0b111 enable Indexed REMAP Mode,
described below*

With all these options it is possible to support in-place transpose,
in-place rotate, Matrix Multiply and Convolutions, without being
limited to Power-of-Two dimension sizes.

## Indexed Mode

Indexed Mode activates reading of the element indices from the GPR
and includes optional limited 2D reordering.
In its simplest form (without elwidth overrides or other modes):

```
    def index_remap(i):
        return GPR((SVSHAPE.SVGPR<<1)+i) + SVSHAPE.offset

    for i in 0..VL-1:
        element_result = ....
        GPR(RT + indexed_remap(i)) = element_result
```

With element-width overrides included, and using the pseudocode
from the SVP64 [[sv/svp64/appendix#elwidth]] elwidth section
this becomes:

```
    def index_remap(i):
        svreg = SVSHAPE.SVGPR << 1
        srcwid = elwid_to_bitwidth(SVSHAPE.elwid)
        offs = SVSHAPE.offset
        return get_polymorphed_reg(svreg, srcwid, i) + offs

    for i in 0..VL-1:
        element_result = ....
        rt_idx = indexed_remap(i)
        set_polymorphed_reg(RT, destwid, rt_idx, element_result)
```

Matrix-style reordering still applies to the indices, except limited
to up to 2 Dimensions (X,Y). Ordering is therefore limited to (X,Y) or
(Y,X) for in-place Transposition.
Only one dimension may optionally be skipped. Inversion of either
X or Y or both is possible (2D mirroring). Pseudocode for Indexed Mode (including elwidth
overrides) may be written in terms of Matrix Mode, specifically
purposed to ensure that the 3rd dimension (Z) has no effect:

```
    def index_remap(ISHAPE, i):
        MSHAPE.skip   = 0b0 || ISHAPE.sk1
        MSHAPE.invxyz = 0b0 || ISHAPE.invxy
        MSHAPE.xdimsz = ISHAPE.xdimsz
        MSHAPE.ydimsz = ISHAPE.ydimsz
        MSHAPE.zdimsz = 0 # disabled
        if ISHAPE.permute = 0b110 # 0,1
           MSHAPE.permute = 0b000 # 0,1,2
        if ISHAPE.permute = 0b111 # 1,0
           MSHAPE.permute = 0b010 # 1,0,2
        el_idx = remap_matrix(MSHAPE, i)
        svreg = ISHAPE.SVGPR << 1
        srcwid = elwid_to_bitwidth(ISHAPE.elwid)
        offs = ISHAPE.offset
        return get_polymorphed_reg(svreg, srcwid, el_idx) + offs
```

The most important observation above is that the Matrix-style
remapping occurs first and the Index lookup second.  Thus it
becomes possible to perform in-place Transpose of Indices which
may have been costly to set up or costly to duplicate
(waste register file space).

-------------

\newpage{}

# svshape instruction  <a name="svshape"> </a>

Form: SVM-Form SV "Matrix" Form (see [[isatables/fields.text]])

    svshape SVxd,SVyd,SVzd,SVRM,vf

| 0.5|6.10  |11.15  |16..20 | 21..24 | 25 | 26..31|  name    |
| -- | --   | ---   | ----- | ------ | -- | ------| -------- |
|OPCD| SVxd | SVyd  | SVzd  | SVRM   | vf | XO    | svshape  |

```
    # for convenience, VL to be calculated and stored in SVSTATE
    vlen <- [0] * 7
    mscale[0:5] <- 0b000001 # for scaling MAXVL
    itercount[0:6] <- [0] * 7
    SVSTATE[0:31] <- [0] * 32
    # only overwrite REMAP if "persistence" is zero
    if (SVSTATE[62] = 0b0) then
        SVSTATE[32:33] <- 0b00
        SVSTATE[34:35] <- 0b00
        SVSTATE[36:37] <- 0b00
        SVSTATE[38:39] <- 0b00
        SVSTATE[40:41] <- 0b00
        SVSTATE[42:46] <- 0b00000
        SVSTATE[62] <- 0b0
        SVSTATE[63] <- 0b0
    # clear out all SVSHAPEs
    SVSHAPE0[0:31] <- [0] * 32
    SVSHAPE1[0:31] <- [0] * 32
    SVSHAPE2[0:31] <- [0] * 32
    SVSHAPE3[0:31] <- [0] * 32

    # set schedule up for multiply
    if (SVrm = 0b0000) then
        # VL in Matrix Multiply is xd*yd*zd
        xd <- (0b00 || SVxd) + 1
        yd <- (0b00 || SVyd) + 1
        zd <- (0b00 || SVzd) + 1
        n <- xd * yd * zd
        vlen[0:6] <- n[14:20]
        # set up template in SVSHAPE0, then copy to 1-3
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[6:11] <- (0b0 || SVyd)   # ydim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim
        SVSHAPE0[28:29] <- 0b11           # skip z
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31]
        SVSHAPE2[0:31] <- SVSHAPE0[0:31]
        SVSHAPE3[0:31] <- SVSHAPE0[0:31]
        # set up FRA
        SVSHAPE1[18:20] <- 0b001          # permute x,z,y
        SVSHAPE1[28:29] <- 0b01           # skip z
        # FRC
        SVSHAPE2[18:20] <- 0b001          # permute x,z,y
        SVSHAPE2[28:29] <- 0b11           # skip y

    # set schedule up for FFT butterfly
    if (SVrm = 0b0001) then
        # calculate O(N log2 N)
        n <- [0] * 3
        do while n < 5
           if SVxd[4-n] = 0 then
               leave
           n <- n + 1
        n <- ((0b0 || SVxd) + 1) * n
        vlen[0:6] <- n[1:7]
        # set up template in SVSHAPE0, then copy to 1-3
        # for FRA and FRT
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D FFT)
        mscale <- (0b0 || SVzd) + 1
        SVSHAPE0[30:31] <- 0b01          # Butterfly mode
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31]
        SVSHAPE2[0:31] <- SVSHAPE0[0:31]
        # set up FRB and FRS
        SVSHAPE1[28:29] <- 0b01           # j+halfstep schedule
        # FRC (coefficients)
        SVSHAPE2[28:29] <- 0b10           # k schedule

    # set schedule up for (i)DCT Inner butterfly
    # SVrm Mode 4 (Mode 12 for iDCT) is for on-the-fly (Vertical-First Mode)
    if ((SVrm = 0b0100) |
        (SVrm = 0b1100)) then
        # calculate O(N log2 N)
        n <- [0] * 3
        do while n < 5
           if SVxd[4-n] = 0 then
               leave
           n <- n + 1
        n <- ((0b0 || SVxd) + 1) * n
        vlen[0:6] <- n[1:7]
        # set up template in SVSHAPE0, then copy to 1-3
        # set up FRB and FRS
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D DCT)
        mscale <- (0b0 || SVzd) + 1
        if (SVrm = 0b1100) then
            SVSHAPE0[30:31] <- 0b11          # iDCT mode
            SVSHAPE0[18:20] <- 0b011         # iDCT Inner Butterfly sub-mode
        else
            SVSHAPE0[30:31] <- 0b01          # DCT mode
            SVSHAPE0[18:20] <- 0b001         # DCT Inner Butterfly sub-mode
            SVSHAPE0[21:23] <- 0b001         # "inverse" on outer loop
        SVSHAPE0[6:11] <- 0b000011       # (i)DCT Inner Butterfly mode 4
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31]
        SVSHAPE2[0:31] <- SVSHAPE0[0:31]
        if (SVrm != 0b0100) & (SVrm != 0b1100) then
            SVSHAPE3[0:31] <- SVSHAPE0[0:31]
        # for FRA and FRT
        SVSHAPE0[28:29] <- 0b01           # j+halfstep schedule
        # for cos coefficient
        SVSHAPE2[28:29] <- 0b10           # ci (k for mode 4) schedule
        SVSHAPE2[12:17] <- 0b000000       # reset costable "striding" to 1
        if (SVrm != 0b0100) & (SVrm != 0b1100) then
            SVSHAPE3[28:29] <- 0b11           # size schedule

    # set schedule up for (i)DCT Outer butterfly
    if (SVrm = 0b0011) | (SVrm = 0b1011) then
        # calculate O(N log2 N) number of outer butterfly overlapping adds
        vlen[0:6] <- [0] * 7
        n <- 0b000
        size <- 0b0000001
        itercount[0:6] <- (0b00 || SVxd) + 0b0000001
        itercount[0:6] <- (0b0 || itercount[0:5])
        do while n < 5
           if SVxd[4-n] = 0 then
               leave
           n <- n + 1
           count <- (itercount - 0b0000001) * size
           vlen[0:6] <- vlen + count[7:13]
           size[0:6] <- (size[1:6] || 0b0)
           itercount[0:6] <- (0b0 || itercount[0:5])
        # set up template in SVSHAPE0, then copy to 1-3
        # set up FRB and FRS
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D DCT)
        mscale <- (0b0 || SVzd) + 1
        if (SVrm = 0b1011) then
            SVSHAPE0[30:31] <- 0b11      # iDCT mode
            SVSHAPE0[18:20] <- 0b011     # iDCT Outer Butterfly sub-mode
            SVSHAPE0[21:23] <- 0b101     # "inverse" on outer and inner loop
        else
            SVSHAPE0[30:31] <- 0b01      # DCT mode
            SVSHAPE0[18:20] <- 0b100     # DCT Outer Butterfly sub-mode
        SVSHAPE0[6:11] <- 0b000010       # DCT Butterfly mode
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31] # j+halfstep schedule
        SVSHAPE2[0:31] <- SVSHAPE0[0:31] # costable coefficients
        # for FRA and FRT
        SVSHAPE1[28:29] <- 0b01           # j+halfstep schedule
        # reset costable "striding" to 1
        SVSHAPE2[12:17] <- 0b000000

    # set schedule up for DCT COS table generation
    if (SVrm = 0b0101) | (SVrm = 0b1101) then
        # calculate O(N log2 N)
        vlen[0:6] <- [0] * 7
        itercount[0:6] <- (0b00 || SVxd) + 0b0000001
        itercount[0:6] <- (0b0 || itercount[0:5])
        n <- [0] * 3
        do while n < 5
           if SVxd[4-n] = 0 then
               leave
           n <- n + 1
           vlen[0:6] <- vlen + itercount
           itercount[0:6] <- (0b0 || itercount[0:5])
        # set up template in SVSHAPE0, then copy to 1-3
        # set up FRB and FRS
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D DCT)
        mscale <- (0b0 || SVzd) + 1
        SVSHAPE0[30:31] <- 0b01          # DCT/FFT mode
        SVSHAPE0[6:11] <- 0b000100       # DCT Inner Butterfly COS-gen mode
        if (SVrm = 0b0101) then
            SVSHAPE0[21:23] <- 0b001     # "inverse" on outer loop for DCT
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31]
        SVSHAPE2[0:31] <- SVSHAPE0[0:31]
        # for cos coefficient
        SVSHAPE1[28:29] <- 0b10           # ci schedule
        SVSHAPE2[28:29] <- 0b11           # size schedule

    # set schedule up for iDCT / DCT inverse of half-swapped ordering
    if (SVrm = 0b0110) | (SVrm = 0b1110) | (SVrm = 0b1111) then
        vlen[0:6] <- (0b00 || SVxd) + 0b0000001
        # set up template in SVSHAPE0
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D DCT)
        mscale <- (0b0 || SVzd) + 1
        if (SVrm = 0b1110) then
            SVSHAPE0[18:20] <- 0b001     # DCT opposite half-swap
        if (SVrm = 0b1111) then
            SVSHAPE0[30:31] <- 0b01          # FFT mode
        else
            SVSHAPE0[30:31] <- 0b11          # DCT mode
        SVSHAPE0[6:11] <- 0b000101       # DCT "half-swap" mode

    # set schedule up for parallel reduction
    if (SVrm = 0b0111) then
        # calculate the total number of operations (brute-force)
        vlen[0:6] <- [0] * 7
        itercount[0:6] <- (0b00 || SVxd) + 0b0000001
        step[0:6] <- 0b0000001
        i[0:6] <- 0b0000000
        do while step <u itercount
            newstep <- step[1:6] || 0b0
            j[0:6] <- 0b0000000
            do while (j+step <u itercount)
                j <- j + newstep
                i <- i + 1
            step <- newstep
        # VL in Parallel-Reduce is the number of operations
        vlen[0:6] <- i
        # set up template in SVSHAPE0, then copy to 1. only 2 needed
        SVSHAPE0[0:5] <- (0b0 || SVxd)   # xdim
        SVSHAPE0[12:17] <- (0b0 || SVzd)   # zdim - "striding" (2D DCT)
        mscale <- (0b0 || SVzd) + 1
        SVSHAPE0[30:31] <- 0b10          # parallel reduce submode
        # copy
        SVSHAPE1[0:31] <- SVSHAPE0[0:31]
        # set up right operand (left operand 28:29 is zero)
        SVSHAPE1[28:29] <- 0b01           # right operand

    # set VL, MVL and Vertical-First
    m[0:12] <- vlen * mscale
    maxvl[0:6] <- m[6:12]
    SVSTATE[0:6] <- maxvl  # MAVXL
    SVSTATE[7:13] <- vlen  # VL
    SVSTATE[63] <- vf
```

Special Registers Altered:

```
    SVSTATE, SVSHAPE0-3
```

`svshape` is a convenience instruction that reduces instruction
count for common usage patterns, particularly Matrix, DCT and FFT. It sets up
(overwrites) all required SVSHAPE SPRs and also modifies SVSTATE
including VL and MAXVL. Using `svshape` therefore does not also
require `setvl`.

Fields:

* **SVxd** - SV REMAP "xdim"
* **SVyd** - SV REMAP "ydim"
* **SVzd** - SV REMAP "zdim"
* **SVRM** - SV REMAP Mode (0b00000 for Matrix, 0b00001 for FFT etc.)
* **vf** - sets "Vertical-First" mode

*Note: SVxd, SVyz and SVzd are all stored "off-by-one".  In the assembler
mnemonic the values `1-32` are stored in binary as `0b00000..0b11111`*

There are 12 REMAP Modes (2 Modes are RESERVED for `svshape2`, 2 Modes
are RESERVED)

| SVRM   | Remap Mode description |
| --     | --              |
| 0b0000 | Matrix 1/2/3D    |
| 0b0001 | FFT Butterfly   |
| 0b0010 | reserved |
| 0b0011 | DCT Outer butterfly  |
| 0b0100 | DCT Inner butterfly, on-the-fly (Vertical-First Mode) |
| 0b0101 | DCT COS table index generation |
| 0b0110 | DCT half-swap   |
| 0b0111 | Parallel Reduction |
| 0b1000 | reserved for svshape2 |
| 0b1001 | reserved for svshape2 |
| 0b1010 | reserved |
| 0b1011 | iDCT Outer butterfly  |
| 0b1100 | iDCT Inner butterfly, on-the-fly (Vertical-First Mode) |
| 0b1101 | iDCT COS table index generation |
| 0b1110 | iDCT half-swap   |
| 0b1111 | FFT half-swap   |

Examples showing how all of these Modes operate exists in the online
[SVP64 unit tests](https://git.libre-soc.org/?p=openpower-isa.git;a=tree;f=src/openpower/decoder/isa;hb=HEAD).  Explaining
these Modes further in detail is beyond the scope of this document.

In Indexed Mode, there are only 5 bits available to specify the GPR
to use, out of 128 GPRs (7 bit numbering).  Therefore, only the top
5 bits are given in the `SVxd` field: the bottom two implicit bits
will be zero (`SVxd || 0b00`).

`svshape` has *limited applicability* due to being a 32-bit instruction.
The full capability of SVSHAPE SPRs may be accessed by directly writing
to SVSHAPE0-3 with `mtspr`. Circumstances include Matrices with dimensions
larger than 32, and in-place Transpose.  Potentially a future v3.1 Prefixed
instruction, `psvshape`, may extend the capability here.

*Architectural Resource Allocation note: the SVRM field is carefully
crafted to allocate two Modes, corresponding to bits 21-23 within the
instruction being set to the value `0b100`, to `svshape2` (not
`svshape`). These two Modes are
considered "RESERVED" within the context of `svshape` but it is
absolutely critical to allocate the exact same pattern in XO for
both instructions in bits 26-31.*

-------------

\newpage{}


# svindex instruction  <a name="svindex"> </a>

SVI-Form

| 0-5|6-10 |11-15  |16-20 | 21-25       | 26-31 |  Form    |
| -- | --  | ---   | ---- | ----------- | ------| -------- |
| PO | SVG | rmm   | SVd  | ew/yx/mm/sk | XO    | SVI-Form |

* svindex SVG,rmm,SVd,ew,SVyx,mm,sk

Pseudo-code:

```
    # based on nearest MAXVL compute other dimension
    MVL <- SVSTATE[0:6]
    d <- [0] * 6
    dim <- SVd+1
    do while d*dim <u ([0]*4 || MVL)
       d <- d + 1

    # set up template, then copy once location identified
    shape <- [0]*32
    shape[30:31] <- 0b00            # mode
    if SVyx = 0 then
        shape[18:20] <- 0b110       # indexed xd/yd
        shape[0:5] <- (0b0 || SVd)  # xdim
        if sk = 0 then shape[6:11] <- 0 # ydim
        else           shape[6:11] <- 0b111111 # ydim max
    else
        shape[18:20] <- 0b111       # indexed yd/xd
        if sk = 1 then shape[6:11] <- 0 # ydim
        else           shape[6:11] <- d-1 # ydim max
        shape[0:5] <- (0b0 || SVd) # ydim
    shape[12:17] <- (0b0 || SVG)        # SVGPR
    shape[28:29] <- ew                  # element-width override
    shape[21] <- sk                     # skip 1st dimension

    # select the mode for updating SVSHAPEs
    SVSTATE[62] <- mm # set or clear persistence
    if mm = 0 then
        # clear out all SVSHAPEs first
        SVSHAPE0[0:31] <- [0] * 32
        SVSHAPE1[0:31] <- [0] * 32
        SVSHAPE2[0:31] <- [0] * 32
        SVSHAPE3[0:31] <- [0] * 32
        SVSTATE[32:41] <- [0] * 10 # clear REMAP.mi/o
        SVSTATE[42:46] <- rmm # rmm exactly REMAP.SVme
        idx <- 0
        for bit = 0 to 4
            if rmm[4-bit] then
                # activate requested shape
                if idx = 0 then SVSHAPE0 <- shape
                if idx = 1 then SVSHAPE1 <- shape
                if idx = 2 then SVSHAPE2 <- shape
                if idx = 3 then SVSHAPE3 <- shape
                SVSTATE[bit*2+32:bit*2+33] <- idx
                # increment shape index, modulo 4
                if idx = 3 then idx <- 0
                else            idx <- idx + 1
    else
        # refined SVSHAPE/REMAP update mode
        bit <- rmm[0:2]
        idx <- rmm[3:4]
        if idx = 0 then SVSHAPE0 <- shape
        if idx = 1 then SVSHAPE1 <- shape
        if idx = 2 then SVSHAPE2 <- shape
        if idx = 3 then SVSHAPE3 <- shape
        SVSTATE[bit*2+32:bit*2+33] <- idx
        SVSTATE[46-bit] <- 1
```

Special Registers Altered:

```
    SVSTATE, SVSHAPE0-3
```

`svindex` is a convenience instruction that reduces instruction count
for Indexed REMAP Mode. It sets up (overwrites) all required SVSHAPE
SPRs and **unlike** `svshape` can modify the REMAP area of the SVSTATE
SPR as well, including setting persistence.  The relevant SPRs *may*
be directly programmed with `mtspr` however it is laborious to do so:
svindex saves instructions covering much of Indexed REMAP capability.

Fields:

* **SVd** - SV REMAP x/y dim 
* **rmm** - REMAP mask: sets remap mi0-2/mo0-1 and SVSHAPEs,
  controlled by mm
* **ew** - sets element width override on the Indices
* **SVG** - GPR SVG<<2 to be used for Indexing
* **yx** - 2D reordering to be used if yx=1
* **mm** - mask mode. determines how `rmm` is interpreted.
* **sk** - Dimension skipping enabled

*Note: SVd, like SVxd, SVyz and SVzd of `svshape`, are all stored
"off-by-one".  In the assembler
mnemonic the values `1-32` are stored in binary as `0b00000..0b11111`*.

*Note: when `yx=1,sk=0` the second dimension is calculated as
`CEIL(MAXVL/SVd)`*.

When `mm=0`:

* `rmm`, like REMAP.SVme, has bit 0 
  correspond to mi0, bit 1 to mi1, bit 2 to mi2,
  bit 3 to mo0 and bit 4 to mi1
* all SVSHAPEs and the REMAP parts of SVSHAPE are first reset (initialised to zero)
* for each bit set in the 5-bit `rmm`, in order, the first
  as-yet-unset SVSHAPE will be updated
  with the other operands in the instruction, and the REMAP
  SPR set.
* If all 5 bits of `rmm` are set then both mi0 and mo1 use SVSHAPE0.
* SVSTATE persistence bit is cleared
* No other alterations to SVSTATE are carried out

Example 1: if rmm=0b00110 then SVSHAPE0 and SVSHAPE1 are set up,
and the REMAP SPR set so that mi1 uses SVSHAPE0 and mi2
uses mi2.  REMAP.SVme is also set to 0b00110, REMAP.mi1=0
(SVSHAPE0) and REMAP.mi2=1 (SVSHAPE1)

Example 2: if rmm=0b10001 then again SVSHAPE0 and SVSHAPE1
are set up, but the REMAP SPR is set so that mi0 uses SVSHAPE0
and mo1 uses SVSHAPE1. REMAP.SVme=0b10001, REMAP.mi0=0, REMAP.mo1=1

Rough algorithmic form:

```
    marray = [mi0, mi1, mi2, mo0, mo1]
    idx = 0
    for bit = 0 to 4:
        if not rmm[bit]: continue
        setup(SVSHAPE[idx])
        SVSTATE{marray[bit]} = idx
        idx = (idx+1) modulo 4
```

When `mm=1`:

* bits 0-2 (MSB0 numbering) of `rmm` indicate an index selecting mi0-mo1
* bits 3-4 (MSB0 numbering) of `rmm` indicate which SVSHAPE 0-3 shall
  be updated
* only the selected SVSHAPE is overwritten
* only the relevant bits in the REMAP area of SVSTATE are updated
* REMAP persistence bit is set.

Example 1: if `rmm`=0b01110 then bits 0-2 (MSB0) are 0b011 and
bits 3-4 are 0b10. thus, mo0 is selected and SVSHAPE2
to be updated. REMAP.SVme[3] will be set high and REMAP.mo0
set to 2 (SVSHAPE2).

Example 2: if `rmm`=0b10011 then bits 0-2 (MSB0) are 0b100 and
bits 3-4 are 0b11.  thus, mo1 is selected and SVSHAPE3
to be updated. REMAP.SVme[4] will be set high and REMAP.mo1
set to 3 (SVSHAPE3).

Rough algorithmic form:

```
    marray = [mi0, mi1, mi2, mo0, mo1]
    bit = rmm[0:2]
    idx = rmm[3:4]
    setup(SVSHAPE[idx])
    SVSTATE{marray[bit]} = idx
    SVSTATE.pst = 1
```

In essence, `mm=0` is intended for use to set as much of the
REMAP State SPRs as practical with a single instruction,
whilst `mm=1` is intended to be a little more refined.

**Usage guidelines**

* **Disable 2D mapping**: to only perform Indexing without
 reordering use `SVd=1,sk=0,yx=0` (or set SVd to a value larger
 or equal to VL)
* **Modulo 1D mapping**: to perform Indexing cycling through the
 first N Indices use `SVd=N,sk=0,yx=0` where `VL>N`. There is
 no requirement to set VL equal to a multiple of N.
* **Modulo 2D transposed**: `SVd=M,sk=0,yx=1`, sets
 `xdim=M,ydim=CEIL(MAXVL/M)`.

Beyond these mappings it becomes necessary to write directly to
the SVSTATE SPRs manually.

-------------

\newpage{}


# svshape2 (offset-priority) <a name="svshape2"> </a>

SVM2-Form

| 0-5|6-9 |10|11-15  |16-20  | 21-24  | 25 | 26-31 |  Form      |
| -- |----|--| ---   | ----- | ------ | -- | ------| --------   |
| PO |offs|yx| rmm   | SVd   | 100/mm | sk | XO    | SVM2-Form  |

* svshape2 offs,yx,rmm,SVd,sk,mm

Pseudo-code:

```
    # based on nearest MAXVL compute other dimension
    MVL <- SVSTATE[0:6]
    d <- [0] * 6
    dim <- SVd+1
    do while d*dim <u ([0]*4 || MVL)
       d <- d + 1
    # set up template, then copy once location identified
    shape <- [0]*32
    shape[30:31] <- 0b00            # mode
    shape[0:5] <- (0b0 || SVd)      # x/ydim
    if SVyx = 0 then
        shape[18:20] <- 0b000       # ordering xd/yd(/zd)
        if sk = 0 then shape[6:11] <- 0 # ydim
        else           shape[6:11] <- 0b111111 # ydim max
    else
        shape[18:20] <- 0b010       # ordering yd/xd(/zd)
        if sk = 1 then shape[6:11] <- 0 # ydim
        else           shape[6:11] <- d-1 # ydim max
    # offset (the prime purpose of this instruction)
    shape[24:27] <- SVo         # offset
    if sk = 1 then shape[28:29] <- 0b01 # skip 1st dimension
    else           shape[28:29] <- 0b00 # no skipping
    # select the mode for updating SVSHAPEs
    SVSTATE[62] <- mm # set or clear persistence
    if mm = 0 then
        # clear out all SVSHAPEs first
        SVSHAPE0[0:31] <- [0] * 32
        SVSHAPE1[0:31] <- [0] * 32
        SVSHAPE2[0:31] <- [0] * 32
        SVSHAPE3[0:31] <- [0] * 32
        SVSTATE[32:41] <- [0] * 10 # clear REMAP.mi/o
        SVSTATE[42:46] <- rmm # rmm exactly REMAP.SVme
        idx <- 0
        for bit = 0 to 4
            if rmm[4-bit] then
                # activate requested shape
                if idx = 0 then SVSHAPE0 <- shape
                if idx = 1 then SVSHAPE1 <- shape
                if idx = 2 then SVSHAPE2 <- shape
                if idx = 3 then SVSHAPE3 <- shape
                SVSTATE[bit*2+32:bit*2+33] <- idx
                # increment shape index, modulo 4
                if idx = 3 then idx <- 0
                else            idx <- idx + 1
    else
        # refined SVSHAPE/REMAP update mode
        bit <- rmm[0:2]
        idx <- rmm[3:4]
        if idx = 0 then SVSHAPE0 <- shape
        if idx = 1 then SVSHAPE1 <- shape
        if idx = 2 then SVSHAPE2 <- shape
        if idx = 3 then SVSHAPE3 <- shape
        SVSTATE[bit*2+32:bit*2+33] <- idx
        SVSTATE[46-bit] <- 1
```

Special Registers Altered:

```
    SVSTATE, SVSHAPE0-3
```

`svshape2` is an additional convenience instruction that prioritises
setting `SVSHAPE.offset`. Its primary purpose is for use when
element-width overrides are used. It has identical capabilities to `svindex` and
in terms of both options (skip, etc.) and ability to activate REMAP
(rmm, mask mode) but unlike `svindex` it does not set GPR REMAP,
only a 1D or 2D `svshape`, and
unlike `svshape` it can set an arbirrary `SVSHAPE.offset` immediate.

One of the limitations of Simple-V is that Vector elements start on the boundary
of the Scalar regfile, which is fine when element-width overrides are not
needed. If the starting point of a Vector with smaller elwidths must begin
in the middle of a register, normally there would be no way to do so except
through LD/ST.  `SVSHAPE.offset` caters for this scenario and `svshape2`is
makes it easier.

**Operand Fields**:

* **offs** (4 bits) - unsigned offset
* **yx** (1 bit) - swap XY to YX
* **SVd** dimension size
* **rmm** REMAP mask
* **mm** mask mode
* **sk** (1 bit) skips 1st dimension if set

Dimensions are calculated exactly as `svindex`. `rmm` and
`mm` are as per `svindex`.  

*Programmer's Note: offsets for `svshape2` may be specified in the range
0-15. Given that the principle of Simple-V is to fit on top of
byte-addressable register files and that GPR and FPR are 64-bit (8 bytes)
it should be clear that the offset may, when `elwidth=8`, begin an
element-level operation starting element zero at any arbitrary byte.
On cursory examination attempting to go beyond the range 0-7 seems
unnecessary given that the **next GPR or FPR** is an
alias for an offset in the range 8-15.  Thus by simply increasing
the starting Vector point of the operation to the next register it
can be seen that the offset of 0-7 would be sufficient.  Unfortunately
however some operations are EXTRA2-encoded it is **not possible**
to increase the GPR/FPR register number by one, because EXTRA2-encoding
of GPR/FPR Vector numbers are restricted to even numbering.
For CR Fields the EXTRA2 encoding is even more sparse.
The additional offset range (8-15) helps overcome these limitations.*

*Hardware Implementor's note: with the offsets only being immediates
and with register numbering being entirely immediate as well it is
possible to correctly compute Register Hazards without requiring
reading the contents of any SPRs.  If however there are
instructions that have directly written to the SVSTATE or SVSHAPE
SPRs and those instructions are still in-flight then this position
is clearly **invalid**. This is why Programmers are strongly
discouraged from directly writing to these SPRs.*

*Architectural Resource Allocation note: this instruction shares
the space of `svshape`. Therefore it is critical that the two
instructions, `svshape` and `svshape2` have the exact same XO
in bits 26 thru 31.  It is also critical that for `svshape2`,
bit 21 of XO is a 1, bit 22 of XO is a 0, and bit 23 of XO is a 0.*

-------------

\newpage{}

# Forms

Add `SVI, SVM, SVRM` to `XO (26:31)` Field in Book I, 1.6.2

Add the following to Book I, 1.6.1, SVI-Form

```
    |0     |6    |11    |16   |21 |23  |24|25|26    31|
    | PO   |  SVG|rmm   | SVd |ew |SVyx|mm|sk|   XO   |
```

Add the following to Book I, 1.6.1, SVM-Form

```
    |0     |6        |11      |16    |21    |25 |26    |31  |
    | PO   |  SVxd   |   SVyd | SVzd | SVrm |vf |   XO      |
```

Add the following to Book I, 1.6.1, SVM2-Form

```
    |0     |6     |10  |11      |16    |21 |24|25 |26    |31  |
    | PO   | SVo  |SVyx|   rmm  | SVd  |XO |mm|sk |   XO      |
```

Add the following to Book I, 1.6.1, SVRM-Form

```
    |0     |6     |11  |13   |15   |17   |19   |21  |22   |26     |31 |
    | PO   | SVme |mi0 | mi1 | mi2 | mo0 | mo1 |pst |///  | XO        |
```

Add the following to Book I, 1.6.2

```
    mi0 (11:12)
        Field used in REMAP to select the SVSHAPE for 1st input register
        Formats: SVRM
    mi1 (13:14)
        Field used in REMAP to select the SVSHAPE for 2nd input register
        Formats: SVRM
    mi2 (15:16)
        Field used in REMAP to select the SVSHAPE for 3rd input register
        Formats: SVRM
    mm (24)
        Field used to specify the meaning of the rmm field for SVI-Form
        and SVM2-Form
        Formats: SVI, SVM2
    mo0 (17:18)
        Field used in REMAP to select the SVSHAPE for 1st output register
        Formats: SVRM
    mo1 (19:20)
        Field used in REMAP to select the SVSHAPE for 2nd output register
        Formats: SVRM
    pst (21)
        Field used in REMAP to indicate "persistence" mode (REMAP
        continues to apply to multiple instructions)
        Formats: SVRM
    rmm (11:15)
        REMAP Mode field for SVI-Form and SVM2-Form
        Formats: SVI, SVM2
    sk (25)
        Field used to specify dimensional skipping in svindex
        Formats: SVI, SVM2
    SVd (16:20)
        Immediate field used to specify the size of the REMAP dimension
        in the svindex and svshape2 instructions
        Formats: SVI, SVM2
    SVDS (16:29)
        Immediate field used to specify a 9-bit signed
        two's complement integer which is concatenated
        on the right with 0b00 and sign-extended to 64 bits.
        Formats: SVDS
    SVG (6:10)
        Field used to specify a GPR to be used as a
        source for indexing.
        Formats: SVI
    SVi (16:22)
         Simple-V immediate field for setting VL or MVL
         Formats: SVL
    SVme (6:10)
         Simple-V "REMAP" map-enable bits (0-4)
         Formats: SVRM
    SVo (6:9)
        Field used by the svshape2 instruction as an offset
        Formats: SVM2
    SVrm (21:24)
         Simple-V "REMAP" Mode
         Formats: SVM
    SVxd (6:10)
         Simple-V "REMAP" x-dimension size
         Formats: SVM
    SVyd (11:15)
         Simple-V "REMAP" y-dimension size
         Formats: SVM
    SVzd (16:20)
         Simple-V "REMAP" z-dimension size
         Formats: SVM
    XO (21:23,26:31)
        Extended opcode field.  Note that bit 21 must be 1, 22 and 23
        must be zero, and bits 26-31 must be exactly the same as
        used for svshape.
        Formats: SVM2
```

# Appendices

    Appendix E Power ISA sorted by opcode
    Appendix F Power ISA sorted by version
    Appendix G Power ISA sorted by Compliancy Subset
    Appendix H Power ISA sorted by mnemonic

| Form | Book | Page | Version | mnemonic | Description |
|------|------|------|---------|----------|-------------|
| SVRM | I    | #    | 3.0B    | svremap    | REMAP enabling instruction |
| SVM  | I    | #    | 3.0B    | svshape  | REMAP shape instruction |
| SVM2 | I    | #    | 3.0B    | svshape2 | REMAP shape instruction (2) |
| SVI  | I    | #    | 3.0B    | svindex | REMAP General-purpose Indexing |

## REMAP pseudocode

Written in python3 the following stand-alone executable source code is the Canonical
Specification for each REMAP. Vectors of "loopends" are returned when Rc=1
in Vectors of CR Fields on `sv.svstep.`, or in Vertical-First Mode
a single CR Field (CR0) on `svstep.`.  The `SVSTATE.srcstep` or `SVSTATE.dststep` sequential
offset is put through each algorithm to determine the actual Element Offset.
Alternative implementations producing different ordering
is prohibited as software will be critically relying on these Deterministic Schedules.

### REMAP 2D/3D Matrix

The following stand-alone executable source code is the Canonical
Specification for Matrix (2D/3D) REMAP. 
Hardware implementations are achievable with simple cascading counter-and-compares.

```
# python "yield" can be iterated. use this to make it clear how
# the indices are generated by using natural-looking nested loops
def iterate_indices(SVSHAPE):
    # get indices to iterate over, in the required order
    xd = SVSHAPE.lims[0]
    yd = SVSHAPE.lims[1]
    zd = SVSHAPE.lims[2]
    # create lists of indices to iterate over in each dimension
    x_r = list(range(xd))
    y_r = list(range(yd))
    z_r = list(range(zd))
    # invert the indices if needed
    if SVSHAPE.invxyz[0]: x_r.reverse()
    if SVSHAPE.invxyz[1]: y_r.reverse()
    if SVSHAPE.invxyz[2]: z_r.reverse()
    # start an infinite (wrapping) loop
    step = 0 # track src/dst step
    while True:
        for z in z_r:   # loop over 1st order dimension
            z_end = z == z_r[-1]
            for y in y_r:       # loop over 2nd order dimension
                y_end = y == y_r[-1]
                for x in x_r:           # loop over 3rd order dimension
                    x_end = x == x_r[-1]
                    # ok work out which order to construct things in.
                    # start by creating a list of tuples of the dimension
                    # and its limit
                    vals = [(SVSHAPE.lims[0], x, "x"),
                            (SVSHAPE.lims[1], y, "y"),
                            (SVSHAPE.lims[2], z, "z")
                           ]
                    # now select those by order.  this allows us to
                    # create schedules for [z][x], [x][y], or [y][z]
                    # for matrix multiply.
                    vals = [vals[SVSHAPE.order[0]],
                            vals[SVSHAPE.order[1]],
                            vals[SVSHAPE.order[2]]
                           ]
                    # ok now we can construct the result, using bits of
                    # "order" to say which ones get stacked on
                    result = 0
                    mult = 1
                    for i in range(3):
                        lim, idx, dbg = vals[i]
                        # some of the dimensions can be "skipped".  the order
                        # was actually selected above on all 3 dimensions,
                        # e.g. [z][x][y] or [y][z][x].  "skip" allows one of
                        # those to be knocked out
                        if SVSHAPE.skip == i+1: continue
                        idx *= mult   # shifts up by previous dimension(s)
                        result += idx # adds on this dimension
                        mult *= lim   # for the next dimension

                    loopends = (x_end |
                               ((y_end and x_end)<<1) |
                                ((y_end and x_end and z_end)<<2))

                    yield result + SVSHAPE.offset, loopends
                    step += 1

def demo():
    # set the dimension sizes here
    xdim = 3
    ydim = 2
    zdim = 4

    # set total (can repeat, e.g. VL=x*y*z*4)
    VL = xdim * ydim * zdim

    # set up an SVSHAPE
    class SVSHAPE:
        pass
    SVSHAPE0 = SVSHAPE()
    SVSHAPE0.lims = [xdim, ydim, zdim]
    SVSHAPE0.order = [1,0,2]  # experiment with different permutations, here
    SVSHAPE0.mode = 0b00
    SVSHAPE0.skip = 0b00
    SVSHAPE0.offset = 0       # experiment with different offset, here
    SVSHAPE0.invxyz = [0,0,0] # inversion if desired

    # enumerate over the iterator function, getting new indices
    for idx, (new_idx, end) in enumerate(iterate_indices(SVSHAPE0)):
        if idx >= VL:
            break
        print ("%d->%d" % (idx, new_idx), "end", bin(end)[2:])

# run the demo
if __name__ == '__main__':
    demo()
```

### REMAP Parallel Reduction pseudocode

The python3 program below is stand-alone executable and is the Canonical Specification
for Parallel Reduction REMAP.
The Algorithm below is not limited to RADIX2 sizes, and Predicate
sources, unlike in Matrix REMAP, apply to the Element Indices **after** REMAP
has been applied, not before.  MV operations are not required: the algorithm
tracks positions of elements that would normally be moved and when applying
an Element Reduction Operation sources the operands from their last-known (tracked)
position.

```
# a "yield" version of the Parallel Reduction REMAP algorithm. 
# the algorithm is in-place. it does not perform "MV" operations.
# instead, where a masked-out value *should* be read from is tracked

def iterate_indices(SVSHAPE, pred=None):
    # get indices to iterate over, in the required order
    xd = SVSHAPE.lims[0]
    # create lists of indices to iterate over in each dimension
    ix = list(range(xd))
    # invert the indices if needed
    if SVSHAPE.invxyz[0]: ix.reverse()
    # start a loop from the lowest step
    step = 1 
    steps = []
    while step < xd:
        step *= 2
        steps.append(step)
    # invert the indices if needed
    if SVSHAPE.invxyz[1]: steps.reverse()
    for step in steps:
        stepend = (step == steps[-1]) # note end of steps
        idxs = list(range(0, xd, step))
        results = []
        for i in idxs:
            other = i + step // 2
            ci = ix[i]
            oi = ix[other] if other < xd else None
            other_pred = other < xd and (pred is None or pred[oi])
            if (pred is None or pred[ci]) and other_pred:
                if SVSHAPE.skip == 0b00: # submode 00
                    result = ci
                elif SVSHAPE.skip == 0b01: # submode 01
                    result = oi
                results.append([result + SVSHAPE.offset, 0])
            elif other_pred:
                ix[i] = oi
        if results:
            results[-1][1] = (stepend<<1) | 1  # notify end of loops
        yield from results

def demo():
    # set the dimension sizes here
    xdim = 9

    # set up an SVSHAPE
    class SVSHAPE:
        pass
    SVSHAPE0 = SVSHAPE()
    SVSHAPE0.lims = [xdim, 0, 0]
    SVSHAPE0.order = [0,1,2]
    SVSHAPE0.mode = 0b10
    SVSHAPE0.skip = 0b00
    SVSHAPE0.offset = 0       # experiment with different offset, here
    SVSHAPE0.invxyz = [0,0,0] # inversion if desired

    SVSHAPE1 = SVSHAPE()
    SVSHAPE1.lims = [xdim, 0, 0]
    SVSHAPE1.order = [0,1,2]
    SVSHAPE1.mode = 0b10
    SVSHAPE1.skip = 0b01
    SVSHAPE1.offset = 0       # experiment with different offset, here
    SVSHAPE1.invxyz = [0,0,0] # inversion if desired

    # enumerate over the iterator function, getting new indices
    shapes = list(iterate_indices(SVSHAPE0)), \
              list(iterate_indices(SVSHAPE1))
    for idx in range(len(shapes[0])):
        l = shapes[0][idx]
        r = shapes[1][idx]
        (l_idx, lend) = l
        (r_idx, rend) = r
        print ("%d->%d:%d" % (idx, l_idx, r_idx),
               "end", bin(lend)[2:], bin(rend)[2:])

# run the demo
if __name__ == '__main__':
    demo()
```

### REMAP FFT pseudocode

The FFT REMAP is RADIX2 only.

```
# a "yield" version of the REMAP algorithm, for FFT Tukey-Cooley schedules
# original code for the FFT Tukey-Cooley schedule:
# https://www.nayuki.io/res/free-small-fft-in-multiple-languages/fft.py
"""
    # Radix-2 decimation-in-time FFT (real, not complex)
    size = 2
    while size <= n:
        halfsize = size // 2
        tablestep = n // size
        for i in range(0, n, size):
            k = 0
            for j in range(i, i + halfsize):
                jh = j+halfsize
                jl = j
                temp1 = vec[jh] * exptable[k]
                temp2 = vec[jl]
                vec[jh] = temp2 - temp1
                vec[jl] = temp2 + temp1
                k += tablestep
        size *= 2
"""

# python "yield" can be iterated. use this to make it clear how
# the indices are generated by using natural-looking nested loops
def iterate_butterfly_indices(SVSHAPE):
    # get indices to iterate over, in the required order
    n = SVSHAPE.lims[0]
    stride = SVSHAPE.lims[2] # stride-multiplier on reg access
    # creating lists of indices to iterate over in each dimension
    # has to be done dynamically, because it depends on the size
    # first, the size-based loop (which can be done statically)
    x_r = []
    size = 2
    while size <= n:
        x_r.append(size)
        size *= 2
    # invert order if requested
    if SVSHAPE.invxyz[0]: x_r.reverse()

    if len(x_r) == 0:
        return

    # start an infinite (wrapping) loop
    skip = 0
    while True:
        for size in x_r:           # loop over 3rd order dimension (size)
            x_end = size == x_r[-1]
            # y_r schedule depends on size
            halfsize = size // 2
            tablestep = n // size
            y_r = []
            for i in range(0, n, size):
                y_r.append(i)
            # invert if requested
            if SVSHAPE.invxyz[1]: y_r.reverse()
            for i in y_r:       # loop over 2nd order dimension
                y_end = i == y_r[-1]
                k_r = []
                j_r = []
                k = 0
                for j in range(i, i+halfsize):
                    k_r.append(k)
                    j_r.append(j)
                    k += tablestep
                # invert if requested
                if SVSHAPE.invxyz[2]: k_r.reverse()
                if SVSHAPE.invxyz[2]: j_r.reverse()
                for j, k in zip(j_r, k_r):   # loop over 1st order dimension
                    z_end = j == j_r[-1]
                    # now depending on MODE return the index
                    if SVSHAPE.skip == 0b00:
                        result = j              # for vec[j]
                    elif SVSHAPE.skip == 0b01:
                        result = j + halfsize   # for vec[j+halfsize]
                    elif SVSHAPE.skip == 0b10:
                        result = k              # for exptable[k]

                    loopends = (z_end |
                               ((y_end and z_end)<<1) |
                                ((y_end and x_end and z_end)<<2))

                    yield (result * stride) + SVSHAPE.offset, loopends

def demo():
    # set the dimension sizes here
    xdim = 8
    ydim = 0 # not needed
    zdim = 1 # stride must be set to 1

    # set total. err don't know how to calculate how many there are...
    # do it manually for now
    VL = 0
    size = 2
    n = xdim
    while size <= n:
        halfsize = size // 2
        tablestep = n // size
        for i in range(0, n, size):
            for j in range(i, i + halfsize):
                VL += 1
        size *= 2

    # set up an SVSHAPE
    class SVSHAPE:
        pass
    # j schedule
    SVSHAPE0 = SVSHAPE()
    SVSHAPE0.lims = [xdim, ydim, zdim]
    SVSHAPE0.order = [0,1,2]  # experiment with different permutations, here
    SVSHAPE0.mode = 0b01
    SVSHAPE0.skip = 0b00
    SVSHAPE0.offset = 0       # experiment with different offset, here
    SVSHAPE0.invxyz = [0,0,0] # inversion if desired
    # j+halfstep schedule
    SVSHAPE1 = SVSHAPE()
    SVSHAPE1.lims = [xdim, ydim, zdim]
    SVSHAPE1.order = [0,1,2]  # experiment with different permutations, here
    SVSHAPE0.mode = 0b01
    SVSHAPE1.skip = 0b01
    SVSHAPE1.offset = 0       # experiment with different offset, here
    SVSHAPE1.invxyz = [0,0,0] # inversion if desired
    # k schedule
    SVSHAPE2 = SVSHAPE()
    SVSHAPE2.lims = [xdim, ydim, zdim]
    SVSHAPE2.order = [0,1,2]  # experiment with different permutations, here
    SVSHAPE0.mode = 0b01
    SVSHAPE2.skip = 0b10
    SVSHAPE2.offset = 0       # experiment with different offset, here
    SVSHAPE2.invxyz = [0,0,0] # inversion if desired

    # enumerate over the iterator function, getting new indices
    schedule = []
    for idx, (jl, jh, k) in enumerate(zip(iterate_butterfly_indices(SVSHAPE0),
                                          iterate_butterfly_indices(SVSHAPE1),
                                          iterate_butterfly_indices(SVSHAPE2))):
        if idx >= VL:
            break
        schedule.append((jl, jh, k))

    # ok now pretty-print the results, with some debug output
    size = 2
    idx = 0
    while size <= n:
        halfsize = size // 2
        tablestep = n // size
        print ("size %d halfsize %d tablestep %d" % \
                (size, halfsize, tablestep))
        for i in range(0, n, size):
            prefix = "i %d\t" % i
            k = 0
            for j in range(i, i + halfsize):
                (jl, je), (jh, he), (ks, ke) = schedule[idx]
                print ("  %-3d\t%s j=%-2d jh=%-2d k=%-2d -> "
                        "j[jl=%-2d] j[jh=%-2d] ex[k=%d]" % \
                                (idx, prefix, j, j+halfsize, k,
                                      jl, jh, ks,
                                ),
                                "end", bin(je)[2:], bin(je)[2:], bin(ke)[2:])
                k += tablestep
                idx += 1
        size *= 2

# run the demo
if __name__ == '__main__':
    demo()
```

### DCT REMAP

DCT REMAP is RADIX2 only.  Convolutions may be applied as usual
to create non-RADIX2 DCT. Combined with appropriate Twin-butterfly
instructions the algorithm below (written in python3), becomes part
of an in-place in-registers Vectorised DCT.  The algorithms work
by loading data such that as the nested loops progress the result
is sorted into correct sequential order.

```
# DCT "REMAP" scheduler to create an in-place iterative DCT.
#

# bits of the integer 'val' of width 'width' are reversed
def reverse_bits(val, width):
    result = 0
    for _ in range(width):
        result = (result << 1) | (val & 1)
        val >>= 1
    return result


# iterative version of [recursively-applied] half-reversing
# turns out this is Gray-Encoding.
def halfrev2(vec, pre_rev=True):
    res = []
    for i in range(len(vec)):
        if pre_rev:
            res.append(vec[i ^ (i>>1)])
        else:
            ri = i
            bl = i.bit_length()
            for ji in range(1, bl):
                ri ^= (i >> ji)
            res.append(vec[ri])
    return res


def iterate_dct_inner_halfswap_loadstore(SVSHAPE):
    # get indices to iterate over, in the required order
    n = SVSHAPE.lims[0]
    mode = SVSHAPE.lims[1]
    stride = SVSHAPE.lims[2]

    # reference list for not needing to do data-swaps, just swap what
    # *indices* are referenced (two levels of indirection at the moment)
    # pre-reverse the data-swap list so that it *ends up* in the order 0123..
    ji = list(range(n))

    levels = n.bit_length() - 1
    ri = [reverse_bits(i, levels) for i in range(n)]

    if SVSHAPE.mode == 0b01: # FFT, bitrev only
        ji = [ji[ri[i]] for i in range(n)]
    elif SVSHAPE.submode2 == 0b001:
        ji = [ji[ri[i]] for i in range(n)]
        ji = halfrev2(ji, True)
    else:
        ji = halfrev2(ji, False)
        ji = [ji[ri[i]] for i in range(n)]

    # invert order if requested
    if SVSHAPE.invxyz[0]:
        ji.reverse()

    for i, jl in enumerate(ji):
        y_end = jl == ji[-1]
        yield jl * stride, (0b111 if y_end else 0b000)

def iterate_dct_inner_costable_indices(SVSHAPE):
    # get indices to iterate over, in the required order
    n = SVSHAPE.lims[0]
    mode = SVSHAPE.lims[1]
    stride = SVSHAPE.lims[2]
    # creating lists of indices to iterate over in each dimension
    # has to be done dynamically, because it depends on the size
    # first, the size-based loop (which can be done statically)
    x_r = []
    size = 2
    while size <= n:
        x_r.append(size)
        size *= 2
    # invert order if requested
    if SVSHAPE.invxyz[0]:
        x_r.reverse()

    if len(x_r) == 0:
        return

    # start an infinite (wrapping) loop
    skip = 0
    z_end = 1 # doesn't exist in this, only 2 loops
    k = 0
    while True:
        for size in x_r:           # loop over 3rd order dimension (size)
            x_end = size == x_r[-1]
            # y_r schedule depends on size
            halfsize = size // 2
            y_r = []
            for i in range(0, n, size):
                y_r.append(i)
            # invert if requested
            if SVSHAPE.invxyz[1]: y_r.reverse()
            # two lists of half-range indices, e.g. j 0123, jr 7654
            j = list(range(0, halfsize))
            # invert if requested
            if SVSHAPE.invxyz[2]: j_r.reverse()
            # loop over 1st order dimension
            for ci, jl in enumerate(j):
                y_end = jl == j[-1]
                # now depending on MODE return the index.  inner butterfly
                if SVSHAPE.skip == 0b00: # in [0b00, 0b10]:
                    result = k  # offset into COS table
                elif SVSHAPE.skip == 0b10: #
                    result = ci # coefficient helper
                elif SVSHAPE.skip == 0b11: #
                    result = size # coefficient helper
                loopends = (z_end |
                           ((y_end and z_end)<<1) |
                            ((y_end and x_end and z_end)<<2))

                yield (result * stride) + SVSHAPE.offset, loopends
                k += 1

def iterate_dct_inner_butterfly_indices(SVSHAPE):
    # get indices to iterate over, in the required order
    n = SVSHAPE.lims[0]
    mode = SVSHAPE.lims[1]
    stride = SVSHAPE.lims[2]
    # creating lists of indices to iterate over in each dimension
    # has to be done dynamically, because it depends on the size
    # first, the size-based loop (which can be done statically)
    x_r = []
    size = 2
    while size <= n:
        x_r.append(size)
        size *= 2
    # invert order if requested
    if SVSHAPE.invxyz[0]:
        x_r.reverse()

    if len(x_r) == 0:
        return

    # reference (read/write) the in-place data in *reverse-bit-order*
    ri = list(range(n))
    if SVSHAPE.submode2 == 0b01:
        levels = n.bit_length() - 1
        ri = [ri[reverse_bits(i, levels)] for i in range(n)]

    # reference list for not needing to do data-swaps, just swap what
    # *indices* are referenced (two levels of indirection at the moment)
    # pre-reverse the data-swap list so that it *ends up* in the order 0123..
    ji = list(range(n))
    inplace_mode = True
    if inplace_mode and SVSHAPE.submode2 == 0b01:
        ji = halfrev2(ji, True)
    if inplace_mode and SVSHAPE.submode2 == 0b11:
        ji = halfrev2(ji, False)

    # start an infinite (wrapping) loop
    while True:
        k = 0
        k_start = 0
        for size in x_r:           # loop over 3rd order dimension (size)
            x_end = size == x_r[-1]
            # y_r schedule depends on size
            halfsize = size // 2
            y_r = []
            for i in range(0, n, size):
                y_r.append(i)
            # invert if requested
            if SVSHAPE.invxyz[1]: y_r.reverse()
            for i in y_r:       # loop over 2nd order dimension
                y_end = i == y_r[-1]
                # two lists of half-range indices, e.g. j 0123, jr 7654
                j = list(range(i, i + halfsize))
                jr = list(range(i+halfsize, i + size))
                jr.reverse()
                # invert if requested
                if SVSHAPE.invxyz[2]:
                    j.reverse()
                    jr.reverse()
                hz2 = halfsize // 2 # zero stops reversing 1-item lists
                # loop over 1st order dimension
                k = k_start
                for ci, (jl, jh) in enumerate(zip(j, jr)):
                    z_end = jl == j[-1]
                    # now depending on MODE return the index.  inner butterfly
                    if SVSHAPE.skip == 0b00: # in [0b00, 0b10]:
                        if SVSHAPE.submode2 == 0b11: # iDCT
                            result = ji[ri[jl]]        # lower half
                        else:
                            result = ri[ji[jl]]        # lower half
                    elif SVSHAPE.skip == 0b01: # in [0b01, 0b11]:
                        if SVSHAPE.submode2 == 0b11: # iDCT
                            result = ji[ri[jl+halfsize]] # upper half
                        else:
                            result = ri[ji[jh]] # upper half
                    elif mode == 4:
                        # COS table pre-generated mode
                        if SVSHAPE.skip == 0b10: #
                            result = k # cos table offset
                    else: # mode 2
                        # COS table generated on-demand ("Vertical-First") mode
                        if SVSHAPE.skip == 0b10: #
                            result = ci # coefficient helper
                        elif SVSHAPE.skip == 0b11: #
                            result = size # coefficient helper
                    loopends = (z_end |
                               ((y_end and z_end)<<1) |
                                ((y_end and x_end and z_end)<<2))

                    yield (result * stride) + SVSHAPE.offset, loopends
                    k += 1

                # now in-place swap
                if inplace_mode:
                    for ci, (jl, jh) in enumerate(zip(j[:hz2], jr[:hz2])):
                        jlh = jl+halfsize
                        tmp1, tmp2 = ji[jlh], ji[jh]
                        ji[jlh], ji[jh] = tmp2, tmp1

            # new k_start point for cos tables( runs inside x_r loop NOT i loop)
            k_start += halfsize


# python "yield" can be iterated. use this to make it clear how
# the indices are generated by using natural-looking nested loops
def iterate_dct_outer_butterfly_indices(SVSHAPE):
    # get indices to iterate over, in the required order
    n = SVSHAPE.lims[0]
    mode = SVSHAPE.lims[1]
    stride = SVSHAPE.lims[2]
    # creating lists of indices to iterate over in each dimension
    # has to be done dynamically, because it depends on the size
    # first, the size-based loop (which can be done statically)
    x_r = []
    size = n // 2
    while size >= 2:
        x_r.append(size)
        size //= 2
    # invert order if requested
    if SVSHAPE.invxyz[0]:
        x_r.reverse()

    if len(x_r) == 0:
        return

    # I-DCT, reference (read/write) the in-place data in *reverse-bit-order*
    ri = list(range(n))
    if SVSHAPE.submode2 in [0b11, 0b01]:
        levels = n.bit_length() - 1
        ri = [ri[reverse_bits(i, levels)] for i in range(n)]

    # reference list for not needing to do data-swaps, just swap what
    # *indices* are referenced (two levels of indirection at the moment)
    # pre-reverse the data-swap list so that it *ends up* in the order 0123..
    ji = list(range(n))
    inplace_mode = False # need the space... SVSHAPE.skip in [0b10, 0b11]
    if SVSHAPE.submode2 == 0b11:
        ji = halfrev2(ji, False)

    # start an infinite (wrapping) loop
    while True:
        k = 0
        k_start = 0
        for size in x_r:           # loop over 3rd order dimension (size)
            halfsize = size//2
            x_end = size == x_r[-1]
            y_r = list(range(0, halfsize))
            # invert if requested
            if SVSHAPE.invxyz[1]: y_r.reverse()
            for i in y_r:       # loop over 2nd order dimension
                y_end = i == y_r[-1]
                # one list to create iterative-sum schedule
                jr = list(range(i+halfsize, i+n-halfsize, size))
                # invert if requested
                if SVSHAPE.invxyz[2]: jr.reverse()
                hz2 = halfsize // 2 # zero stops reversing 1-item lists
                k = k_start
                for ci, jh in enumerate(jr):   # loop over 1st order dimension
                    z_end = jh == jr[-1]
                    if mode == 4:
                        # COS table pre-generated mode
                        if SVSHAPE.skip == 0b00: # in [0b00, 0b10]:
                            if SVSHAPE.submode2 == 0b11: # iDCT
                                result = ji[ri[jh]] # upper half
                            else:
                                result = ri[ji[jh]]        # lower half
                        elif SVSHAPE.skip == 0b01: # in [0b01, 0b11]:
                            if SVSHAPE.submode2 == 0b11: # iDCT
                                result = ji[ri[jh+size]] # upper half
                            else:
                                result = ri[ji[jh+size]] # upper half
                        elif SVSHAPE.skip == 0b10: #
                            result = k # cos table offset
                    else:
                        # COS table generated on-demand ("Vertical-First") mode
                        if SVSHAPE.skip == 0b00: # in [0b00, 0b10]:
                            if SVSHAPE.submode2 == 0b11: # iDCT
                                result = ji[ri[jh]]        # lower half
                            else:
                                result = ri[ji[jh]]        # lower half
                        elif SVSHAPE.skip == 0b01: # in [0b01, 0b11]:
                            if SVSHAPE.submode2 == 0b11: # iDCT
                                result = ji[ri[jh+size]] # upper half
                            else:
                                result = ri[ji[jh+size]] # upper half
                        elif SVSHAPE.skip == 0b10: #
                            result = ci # coefficient helper
                        elif SVSHAPE.skip == 0b11: #
                            result = size # coefficient helper
                    loopends = (z_end |
                               ((y_end and z_end)<<1) |
                                ((y_end and x_end and z_end)<<2))

                    yield (result * stride) + SVSHAPE.offset, loopends
                    k += 1

            # new k_start point for cos tables( runs inside x_r loop NOT i loop)
            k_start += halfsize

```

## REMAP selector

Selecting which REMAP Schedule to use is shown by the pseudocode below.
Each SVSHAPE 0-3 goes through this selection process.

```
    if SVSHAPEn.mode == 0b00:
        iterate_fn = iterate_indices
    elif SVSHAPEn.mode == 0b10:
        iterate_fn = iterate_preduce_indices
    elif SVSHAPEn.mode in [0b01, 0b11]:
        # further sub-selection
        if SVSHAPEn.ydimsz == 1:
            iterate_fn = iterate_butterfly_indices
        elif SVSHAPEn.ydimsz == 2:
            iterate_fn = iterate_dct_inner_butterfly_indices
        elif SVSHAPEn.ydimsz == 3:
            iterate_fn = iterate_dct_outer_butterfly_indices
        elif SVSHAPEn.ydimsz in [5, 13]:
            iterate_fn = iterate_dct_inner_costable_indices
        elif SVSHAPEn.ydimsz in [6, 14, 15]:
            iterate_fn = iterate_dct_inner_halfswap_loadstore
```


[[!tag opf_rfc]]